analytical expression of confinement loss in bragg fibers and ......the antiresonant reflecting...

Analytical expression of confinement loss in Bragg fibersand its relationship with generalized

quarter-wave stack condition

Jun-ichi Sakai

Faculty of Science and Engineering, Ritsumeikan University, Noji-higashi, Kusatsu-city, Shiga 525-8577, Japan([email protected])

Received May 17, 2011; revised August 30, 2011; accepted September 17, 2011;posted September 22, 2011 (Doc. ID 147654); published October 24, 2011

Analytical expression of confinement loss (CL) is presented for the TE mode of Bragg fibers by applying the per-turbation theory to electromagnetic fields under an asymptotic expansion approximation. The expression can beused to evaluate CL for general cases as well as the quarter-wave stack (QWS) condition. Under the generalizedQWS condition, the expression gives an explicit dependence of CL on the fiber structural parameters, the numberof periodic layer pairs, and so on. We also present numerical results for dependence of the CL on various param-eters, such as the core radius, the cladding layer thicknesses, the refractive indices, the wavelengths, the number ofperiodic layer pairs, and the mode number. The relation between the CL and the generalized QWS condition isscrutinized. The position of the minimum CL corresponds well to the condition satisfying a generalized QWScondition. © 2011 Optical Society of America

OCIS codes: 060.2270, 060.2310, 060.2400.

1. INTRODUCTIONThe periodicity of cladding in photonic crystal fibers (PCFs)helps in optical confinement due to the photonic bandgap.However, the finite thickness of the periodic cladding causesan optical leakage, and this phenomenon is called the confine-ment loss (CL) [1]. Reduction in CL is important for ap-propriate design of PCFs because CL can arise even withoutmaterial and scattering losses.

CL is usually evaluated by numerical means. An index-guid-ing holey fiber, which includes air holes in the cladding, makesuse of the multipole method [1,2]. A hollow-core photonicbandgap fiber (PBF) with air holes in the cladding utilizes thevectorial finite element method [3]. In an all-solid PBF basedon the antiresonant reflection, the adjustable boundary con-dition method is used [4].

CL of a Bragg fiber [5,6], which consists of air core and cy-lindrically symmetric cladding with alternating high and lowrefractive indices, has been studied by the transfer matrixmethod [7,8], Chew’s method [9,10], a mixed method of exactand asymptotic analyses [11], a multilayer division (MLD)method [12], and the finite element method [13]. The com-bined effect of confinement and material losses in the Braggfiber has been investigated numerically [11,14]. Although nu-merical means give detailed numerical results, they not onlyrequire much computing time but also are disadvantageous towholely grasping the characteristics.

An analytical approach to characterizing the Bragg fiber isan asymptotic expansion approximation [15–18], where elec-tromagnetic fields are exactly treated for the core and aretreated in an asymptotic form for the cladding. Use of approx-imate Bloch theorem in the cylindrical coordinates improvesthe accuracy of eigenvalue equations especially for hybridmodes [19].

Enhancement of the optical confinement to the core is rea-lized by the introduction of quarter-wave stack (QWS) condi-tion [5]. Its employment vastly simplifies the eigenvalueequation from the viewpoint of wave [18] and geometrical op-tics [20]. Relation of QWS condition with the optimal indexprofile has been discussed [21].

The antiresonant reflecting optical waveguide (ARROW)model was presented to confine optical waves in the wave-guide with high-index single cladding [22]. The model hasbeen generalized to optical waveguides and fibers with manycladding layers by counting the phase at each round trip with-in individual layers [23]. The ARROW model has been ex-tended to a stratified planar ARROW (SPARROW) model [13].A phase theory has shown the equivalence between the in-phase condition at the core–cladding boundary and theantiresonant reflection condition at each round trip in theone-dimensional (1D) and cylindrically symmetric two-dimensional (2D) structures with periodic cladding [24]. Italso shows that a generalized QWS condition developed fromthe QWS condition is equivalent to the central gap point in theSPARROW model.

Evaluation of the CL requires the imaginary part of thecomplex propagation constant. It is generally cumbersometo numerically solve the eigenvalue equation, including thecomplex propagation constant in the Bragg fiber. The firstpurpose of the present paper is to present and verify an ana-lytical expression of CL for the TE mode, the lowest CL mode[7,12], of Bragg fibers with finite periodic region to give phy-sical insight into the PBFs. The CL is evaluated by treating thefinite periodic cladding as a perturbation from the Bragg fiberwith infinite periodic cladding [18], which is analyzed using anasymptotic expansion approximation. The second purpose isto relate the CL characteristics with the generalized QWS con-dition and to find the operation condition showing the low CL.

2740 J. Opt. Soc. Am. B / Vol. 28, No. 11 / November 2011 Jun-ichi Sakai

0740-3224/11/112740-15$15.00/0 © 2011 Optical Society of America

This paper proceeds as follows. Section 2 shows how toderive the propagation constant in the perturbed system,especially for a Bragg fiber. Section 3 derives perturbed fieldsin the Bragg fiber. Section 4 provides the relationship betweenamplitude coefficients of perturbed fields in restricted re-gions. Amplitude coefficients of perturbed fields are deter-mined in Section 5. In Section 6, we present an analyticalexpression of the propagation constant in the perturbed sys-tem. Section 7 gives an explicit expression of the propagationconstant under the QWS condition. Section 8 compares thepresent results with those of the MLD method to verify thepresent perturbation theory. Section 9 clarifies its relationshipwith the generalized QWS condition. In addition, we show thedesign method of a low CL. Sections 10 and 11 are devoted todiscussion and conclusion.

2. BASIC EQUATIONS BASED ONPERTURBATION THEORY

A. General Expressions of Propagation ConstantWe use cylindrical coordinates �r; θ; z� with the optical propa-gation axis z. Assume that the waveguide structure is indepen-dent of z. Electromagnetic field components are assumed tohave a spatiotemporal factor of Utz � exp�i�ωt − βz�� with theangular frequency ω and the complex propagation constant β,and the factor is dropped off hereafter. The CL is calculatedfrom the imaginary part Im�β� of propagation constant β and itis required that Im�β� < 0 in the present framework.

The electromagnetic component Ψ satisfies the waveequation

�∇2t � ϵ�r�k20 − β2�Ψ � 0; �1�

where ϵ�r� denotes the dielectric constant consisting of a stepfunction, k0 � 2π=λ0 denotes the wavenumber of vacuum, andλ0 denotes the wavelength of vacuum. CL is calculated here bytreating the present system as a perturbation from the Braggfiber with infinite periodic cladding, which has the same fiberparameters as those of the perturbed system, except that theperiodic cladding extends to infinity. In this case, the changein dielectric constant can be represented by

ϵ�r� � ϵ∞�r� � δϵp�r�: �2�

Here, δ is a parameter representing the order of perturbation,and subscripts ∞ and p indicate parameters of unperturbedand perturbed systems.

To solve Eq. (1) in the perturbed system, fields and propa-gation constant are set as

Ψ � Ψ∞ � δΨp1 � δ2Ψp2; �3a�

β � β∞ � δβp1 � δ2βp2: �3b�

Substitution of these equations into Eq. (1) yields

�∇2t � ϵ∞�r�k20 − β2∞�Ψp1 � �−ϵp�r�k20 � 2β∞βp1�Ψ∞ �4�

for the first-order perturbation and

�∇2t � ϵ∞�r�k20 − β2∞�Ψp2 � �−ϵp�r�k20 � 2β∞βp1�Ψp1� �2β∞βp2 � β2p1�Ψ∞ �5�

for the second-order perturbation. Equations (4) and (5) holdeven when Ψ is replaced with a scalar function.

We first dot multiply both sides of Eq. (5) by Ψ�∞ and inte-grate the entire fiber cross section. Moreover, we apply theGreen theorem to the result and employ Ψ�∞ � ∇t ·Ψ�∞ � 0at r � ∞. Then we have

ZZΨ�∞ ·∇2tΨp1dS �

ZZΨp1 ·∇2tΨ�∞dS �6�

through the standard procedure. In the case of real β∞, whichcan be applied to the Bragg fiber with infinite periodic clad-ding, we finally have the propagation constant βp1 of the first-order perturbation as

βp1 �RR

ϵp�r�k20∣Ψ∞∣2dS2β∞

RR∣Ψ∞∣2dS

: �7�

It can be seen from Eq. (7) that βp1 is real for a real value ofϵp�r�. We need a higher-order perturbation term to evaluatethe CL.

In the same way as with βp1, we can obtain the second-order perturbation of the propagation constant from Eq. (5) as

βp2�−β2p1

RR∣Ψ∞∣2dS�

RR �ϵp�r�k20 −2β∞βp1�Ψ�∞ ·Ψp1dS2β∞

RR∣Ψ∞∣2dS

: �8�

Equation (8) implies that βp2 can be evaluated using the first-order perturbation functions, Ψp1 and βp1, and unperturbedfunctions, Ψ∞ and β∞.

B. Formal Expressions of CL for Bragg FibersConsider a Bragg fiber with finite periodic cladding and a cy-lindrically symmetric index (see Fig. 1). The fiber consists of acore, periodic cladding, and infinite external region. The coreindex is nc and its radius is rc. The periodic cladding has afinite number N of layer pairs that consist of high na and

Fig. 1. (Color online) Schematics of Bragg fiber: rc, core radius; nc,refractive index of core; na and nb, refractive indices of layers withthicknesses a and b, respectively (na > nb > nc);Λ � a� b, period inperiodic region; nex, refractive index of external region; N , number ofperiodic cladding pairs; rmA ≡ rc � �m − 1�Λ� a; rmB ≡ rc �mΛ; Acand Ap1;c, amplitude coefficients of unperturbed and perturbed fields,respectively, for the core; am and bm, amplitude coefficients of unper-turbed fields formth cladding layer a; ap1;m and bp1;m, amplitude coef-ficients of perturbed fields for mth cladding layer a. Amplitudecoefficients formth cladding layer b are indicated by adding the primeto those for cladding layer a. Dotted lines indicate an unperturbedsystem in which the periodic structure extends to infinity.

Jun-ichi Sakai Vol. 28, No. 11 / November 2011 / J. Opt. Soc. Am. B 2741

low indices nb (na > nb > nc). The corresponding layer thick-nesses are a and b, and the cladding period is Λ � a� b. Theindex of the external region is denoted by nex and is set tonex � nb in the present treatment.

CL is calculated here by treating the present system as aperturbation from the Bragg fiber with infinite periodic clad-ding [18], which has the core and periodic cladding para-meters identical with those of the perturbed system.Namely, the fiber with infinite periodic cladding is regardedas an unperturbed system. Then, the perturbed dielectric con-stant can be represented by

ϵp�r��−�n2a −n2b�; rc�mΛ ≤ r ≤ rc�mΛ�a �m�N −∞�

0; otherwise:

�9�

Applying Eq. (9) to Eq. (7), we obtain a formal expression ofthe propagation constant of the first-order perturbation as

βp1 � −�n2a − n2b�k20Iex;a

2β∞Itot; �10�

where Itot denotes the total power and is defined by

Itot ≡ZZ

∣Ψ∞∣2dS � Icore � I∞a � I∞b: �11�

Here, parameters Icore, I∞a, and I∞b indicate integrals overthe core, entire periodic cladding layer a, and entire periodiccladding layer b, respectively, in the infinite periodic claddingcase. Iex;a is an integral value of ∬∣Ψ∞∣2dS over the entire ex-ternal layer a.

For the second term of the numerator of Eq. (8), we have

Ip2 ≡ZZ

�ϵp�r�k20 − 2β∞βp1�Ψ�∞ ·Ψp1dS

� −2β∞βp1�Ip;core � Ip;pe;a � Ip;pe;b � Ip;ex;b�− ��n2a − n2b�k20 � 2β∞βp1�Ip;ex;a; �12�

where integral parameters, Ip;core, Ip;pe;a, Ip;pe;b, Ip;ex;a, andIp;ex;b, indicate the contribution of the core, periodic claddinglayers a and b and external layers a and b, respectively.

3. DERIVATION OF PERTURBED FIELDS INBRAGG FIBERFields of unperturbed Bragg fiber that has an infinite periodiccladding are represented by exact fields for the core and byfields analyzed using an asymptotic expansion approximationfor the cladding [18]. A Bragg fiber with the finite periodic re-gion is treated as the perturbation from the unperturbed Braggfiber. Nonzero electromagnetic fields to be continuous at theinterfaces are Hz and Eθ for the TE mode.

A. Unperturbed FieldsFor a fiber with infinite periodic cladding, nonzero electro-magnetic fields for Ψ∞ are expressed in a matrix form as [18]

�Hz;∞iEθ;∞

��

8>><>>:

Dc�r��AcBc

�for the core

Di�r�a∞;m; a∞;m ≡�ambm

�for the cladding

;

�13�

Dc�r�≡� d11 d12d21 d22

�;

Di�r�≡�2πr

�1=2

Gi�r�Qi�r� �i � a; b�: �14a�

Here, Ac and Bc denote the amplitude coefficients of the core,and a∞;m denotes the column vector representing the ampli-tude coefficients, am and bm, of the mth cladding layer a(Fig. 1). Those for cladding layer b are expressed by addingthe prime. These amplitude coefficients apply to the unper-turbed fiber. We set Bc � Ac from a requirement that fieldsmust be finite at the core center.

Elements of representation matrix Dc�r� in the core aregiven by

d11 � H�2�ν �κcr�; d12 � H�1�ν �κcr�;d21 � −

ωμ0κc

H�2�0ν �κcr�; d22 � −ωμ0κc

H�1�0ν �κcr� �14b�

with the lateral propagation constant

κi � ��nik0�2 − β2∞�1=2 �i � a; b; c�: �15�

H�1�ν � Jν � iNν andH�2�ν � Jν − iNν indicate the Hankel func-tions of the first and second kinds, respectively, ν denotes theazimuthal mode number, and μ0 denotes the magnetic perme-ability of vacuum. Although ν � 0 for the TE mode, it isretained for reference. The prime of cylinder functions indi-cates differentiation with respect to their argument through-out this paper.

The boundary matrix Gi�r� and the displacement matrixQi�r� included in the cladding representation are expressed as

Gi�r� �1��κip

�1 1

iωμ0=κi −iωμ0=κi

��i � a; b�; �16a�

Qi�r� ��exp�−iκir� 0

0 exp�iκir��

�16b�

under the asymptotic expansion approximation, which is validonly for κir ≫ 1. Functions exp�−iκir� and exp�iκir� corre-spond to outward and inward waves, respectively. Relativeradial coordinates, ra;m and rb;m, are introduced to calculatethe phase factor of the cladding layers a and b, respectively,where ra;m ≡ r − �rc � �m − 1�Λ�, rb;m ≡ r − �rc � �m − 1�Λ�a�, 0 ≤ ra;m ≤ a, and 0 ≤ rb;m ≤ b.

After the propagation constant β∞ is determined throughsolving eigenvalue equations, amplitude coefficients, am,bm, a0m, and b0m, of cladding layers are represented in termsof amplitude coefficients a1 and b1 or Ac [17,18]. Some impor-tant properties in deriving equations of the periodic structureare summarized in Appendix A.


B. Perturbed FieldsWe use a parameter variation method to deriveHz;p1�r� from adifferential equation, where Ψ is set as Hz�r� in Eq. (4).iEθ;p1�r� can be derived from Hz;∞�r� and Hz;p1�r�. Perturbedfields are summarized in terms of matrix form. For the core,we have

�Hz;p1�r�iEθ;p1�r�

�� Dc�r�

�Ap1;cBp1;c

�−β∞βp1κc

rMc�r��AcBc

�: �17a�

Here, Ap1;c and Bp1;c stand for the amplitude coefficients of theperturbed system. Matrix elements mij of Mc�r� are given by

m11 � H�2�0ν �κcr�; m12 � H�1�0ν �κcr�;

m21 �ωμ0κc

��1 −

ν2κ2cr2

�H�2�ν �κcr� �

2κcr

H�2�0ν �κcr��;

m22 �ωμ0κc

��1 −

ν2κ2cr2

�H�1�ν �κcr� �

2κcr

H�1�0ν �κcr��: �17b�

Similarly, one obtains perturbed fields of mth periodic andexternal layers a as

�Hz;p1�r�iEθ;p1�r�

��

�Di�r�ape;m�r� for periodic layersDi�r�aex;m�r� for external layers ; �18�

where two column vectors are defined by

ape;m�r� �� ap1;mbp1;m

�− i

β∞βp1κi

r�−am

bm

�

�1 ≤ m ≤ N; i � a; b�; �19�

aex;m�r��ap1;mbp1;m

�−iηpir

�−ambm

��m≥N�1;i�a;b�; �20�

ηpi ≡12κi

��n2i − n2b�k20 � 2β∞βp1� �i � a; b�: �21�

Here, ap1;m and bp1;m stand for the amplitude coefficients ofthe mth layer a (see Fig. 1). The β∞βp1=κi and ηpi in Eqs. (19)and (20) indicate effects of the propagation constant and re-fractive index changes due to the finite thickness of the per-iodic layer, respectively. Actually, ηpb is equal to β∞βp1=κb. Inηpa of Eq. (21), the first is much larger than the second, asshown in the last paragraph of Subsection 7.B. The amplitudecoefficients for mth layers b are expressed by adding theprime to column vectors ape;m�r� and aex;m�r�.

Equation (18) indicates that the relationship between am-plitude coefficients in the perturbed system is identical to thatin the unperturbed system if perturbed fields are defined byexpressions in Eqs. (19) and (20). This property is peculiar tothe asymptotic expansion approximation and useful in that ifape;m�r� or a0pe;m�r� corresponds to the column vector given inEq. (A1), then properties similar to those of the unperturbedsystem can be used in the periodic region.

4. RELATIONSHIP BETWEEN AMPLITUDECOEFFICIENTS OF PERTURBED FIELDSThere are many amplitude coefficients to be related with eachother in the perturbed fields. The relationship between theamplitude coefficients must be treated systematically. Condi-tions of amplitude coefficients are summarized in thefollowing way: (i) fields must be finite at the core center;(ii) tangential components must be continuous at the interfaceof the core, periodic layers, and external layers; and (iii) in-ward field components must disappear in the external layers.The last condition is peculiar to the present problem.

A. Amplitude Coefficients in Core and Periodic RegionFrom a requirement that fields must be finite at the core cen-ter, one obtains Ap1;c � Bp1;c because the unperturbed fieldshave been set to be finite at the core center.

For perturbed fields between themth periodic layers a andb, we require

Ga�rmA�Qa�ra;m � a�ape;m�rmA�� Gb�rmA�Qb�rb;m � 0�a0pe;m�rmA� �22�

with rmA ≡ rc � �m − 1�Λ� a (see Fig. 1). Here, Qb�rb;m � 0�is a unit matrix. Multiplication of the above equation byG−1b �rmA� from the left-hand side produces the relation

a0pe;m�rmA� � Haape;m�rmA� �1 ≤ m ≤ N�; �23�

where

Ha � G−1b �rmA�Ga�rmA�Qa�ra;m � a� �12

�κbκa

�1=2

fhaijg: �24�

Matrix elements of Ha are the same as those of the unper-turbed system: ha11 � ha�22 � �1� κb=κa� exp�−iκaa� and ha12 �ha�21 � �1 − κb=κa� exp�iκaa�. Equation (23) indicates the ampli-tude coefficients of cladding layer b in terms of those of clad-ding layer a, and the left-hand side is symmetrical to the right-hand side regarding matrix Ha.

Amplitude coefficients of (m� 1)th cladding layer a are si-milarly related with those of the mth cladding layer b atr � rmB ≡ rc �mΛ; that is, ape;m�1�rmB� � Hba0pe;m�rmB�.Here, we define Hb � �1=2��κa=κb�1=2fhbijg, the parametersof which are obtained by interchanging a and b in fhaijg.

The relationship between amplitude coefficients in adja-cent layers a of the periodic region can be written by

�ap1;m�1bp1;m�1

�� exp�−iKjΛ�

�ap1;mbp1;m

�

� iβ∞βp1b exp�−iKjmΛ��1κa

�−a1b1

�

−1κb

exp�iKjΛ�Hb�−a01b01

��25�

using the above two relations and a property concerningHbHa (see Appendix A) with Kj being the Bloch wavenumber(j � 1; 2).

Summation of both sides of Eq. (25) multiplied byexp�−iKj�m − 1 −m0�Λ� overm0 � 1 tom − 1 yields amplitudecoefficients of the mth periodic layer a as


�ap1;mbp1;m

�� exp�−iKj�m − 1�Λ�

��ap1;1bp1;1

�

� iβ∞βp1�m − 1�b�1κa

�−a1b1

�

−1κb


��: �26�

Equation (26) indicates that the amplitude coefficients of theperturbed system can be represented using those of the firstcladding layer in perturbed and unperturbed systems.

Combination of Eq. (23) with Eq. (26) produces similar am-plitude coefficients of the mth periodic layer b as

�a0p1;mb0p1;m

�� exp�−iKj�m − 1�Λ�

�Ha

�ap1;1bp1;1

�

� iβ∞βp1�rc �ma��−1κa

Ha

�−a1b1

�

� 1κb

�−a01b01

��: �27�

B. Amplitude Coefficients and Fields for External RegionField Ψ∞ �Ψp1 in the external region can be represented by

a0∞;m � a0ex;m�r� � Ha�a∞;m � aex;m�r�� m ≥ N � 1�: �28�

Inward wave components must disappear in the form ofΨ∞ �Ψp1 because the external region has no periodic structure.This requirement results in

bm � bp1;m − iηparbm � 0; b0m � b0p1;m − iηpbrb0m � 0�m ≥ N � 1�: �29�

Outward waves are continuous at r � rmA and rmB in theexternal region. The relationship between amplitude coeffi-cients in adjacent layers a of the external region becomes

ap1;m�1 � Cexap1;m − �1� iηparmB�a1 exp�−iKjmΛ�� Cex�1� iηparmA�a1 exp�−iKj�m − 1�Λ� �30�

with

Cex ≡14

�2� κbκa

� κaκb

�exp�−i�κaa� κbb��: �31�

Cex represents an effect that outward waves propagatethrough the external region without reflection at the layer in-terface. It should be noted that HbHa is not replaced byexp�−iKjΛ� in the external region. Summation of both sidesof Eq. (30) multiplied by C�N�q−1�−mex over m � N � 1 to N �q − 1 leads to the amplitude coefficient of the qth externallayer a as

ap1;N�q � ap1;N�1Cq−1ex − aN�1 exp�−iKj�q − 1�Λ�

×�1� iηparN�q−1;B −

iηpaaCex exp�iKjΛ�1 − Cex exp�iKjΛ�

�

� aN�1Cq−1ex�1� iηparN�1;A −

iηpaa1 − Cex exp�iKjΛ�

�:

�32�

In a similar way, we have the amplitude coefficient of theqth external layer b as

a0p1;N�q�ap1;N�1Cq−1ex12

�κbκa

�1=2

�1� κbκa

�exp�−iκaa�

�aN�1Cq−1ex �1� iηparN�1;A�12

�κbκa

�1=2

×�1� κbκa


�a0N�1fexp�−iKj�q−1�Λ�−Cq−1ex giηpbbCex exp�iKjΛ�1−Cex exp�iKjΛ�

−a0N�1�1� iηpbrN�q;A�exp�−iKj�q−1�Λ�: �33�

Equations (32) and (33) mean that amplitude coefficients ofthe (N � q)th layers a and b in the perturbed system are ex-pressible as a function of amplitude coefficients of the firstexternal layer in the perturbed and unperturbed systems.

Perturbed fields of the qth external layer a are expressed as

�Hz;p1iEθ;p1

�� Da�r�

��aN�q � ap1;N�q � iηparaN�q

0

�− a∞;N�q

�

� exp�−iKj�q − 1�Λ�Da�r��iηpar

�aN�10

�

−�

0bN�1

��

�a†N�q0

�� Cq−1ex Da�r�

�a†p1;N�1

0

�;

�34�

where we used Eq. (32) and set

a†N�q≡−aN�1�1�iηparN�q−1;B−

iηpaaCex exp�iKjΛ�1−Cex exp�iKjΛ�

�; �35a�

a†p1;N�1 ≡ ap1;N�1

� aN�1�1� iηparN�1;A −

iηpaa1 − Cex exp�iKjΛ�

�: �35b�

In Eq. (34), the second term takes place to cancel out the in-ward wave in unperturbed fields, and the remaining termsconsist of only the outward wave component. The first termincludes the radial coordinate r. Here, a†N�q depends on q, buta†p1;N�1 is independent of q.

5. DETERMINATION OF AMPLITUDECOEFFICIENTS IN PERTURBED FIELDSAmplitude coefficients between the core and innermost exter-nal layer in the perturbed system can be related with eachother with the help of the relationship among amplitude co-efficients in the periodic region. Through this process, allthe amplitude coefficients in the perturbed system can be


explicitly expressed in terms of known parameters of the un-perturbed system.

Only the perturbed fields must be continuous at the inter-face of the core and innermost periodic layer since the unper-turbed fields have been set to be continuous there. This isaccomplished by applying the boundary condition toEqs. (17a) and (18) at r � rc and by utilizing the finite fieldrequirement at the core center. In the external region, onlythe outward fields exist in the form of Ψ∞ �Ψp1, as shownin Eq. (29). From this condition, fields are related at the inter-face of the outermost periodic and innermost external layers,namely r � rNB. Amplitude coefficients, Ap1;c and ap1;N�1, aretreated here as independent parameters. Thus, we have a re-lationship of amplitude coefficients between the core and in-nermost external layer as

�J�;a − exp�iKjNΛ�J−;a 0

��Ap1;c�πκarc=2�1=2

ap1;N�1

��

�PaPb

�;

�36a�

where

�PaPb

�≡ iηparNB

�a10

�− β∞βp1

rcκc

FcAc

�πκarc2

�1=2

−�

0b1

�

� iβ∞βp1Naκa

�−a1b1

�� iβ∞βp1

Nbκb


�;

�36b�

J�;i ≡ Jν � iκiκcJ 0ν �i � a; b�; �36c�

Fc ≡

0BBB@

−J 0ν � i κaκc��

1 − ν2κ2c r2c

�Jν � 2J 0ν

�

−�J 0ν � i κaκc

��1 − ν2κ2c r2c

�Jν � 2J 0ν

��1CCCA: �36d�

In the double sign notation of Eq. (36c), upper and lower signscorrespond to each other. The argument κcrc of Jν and J 0ν isdropped off here. The product HbHa was replaced byexp�−iKjΛ� in the periodic region (see Appendix A).

Equation (36a) enables us to solve the amplitude coeffi-cients, Ap1;c and ap1;N�1, of the perturbed system in termsof structural parameters and amplitude coefficients of the un-perturbed system. Note that βp1 can be evaluated using param-eters of the unperturbed system, as shown in Eq. (7) orEq. (10). In Eq. (36b), the first term indicates the effect of in-dex change from na to nb in the external layer, the secondindicates an influence on the core, the third indicates the ef-fect of no inward wave in the external layer, the fourth indi-cates the phase change arising from all periodic layers a, andthe last term indicates the phase change caused by all periodiclayers b.

Solving Eq. (36a), we obtain the amplitude coefficients ofthe perturbed system (see Appendix B). We take into consid-eration that terms being proportional to β∞βp1 are small com-pared to other terms. Then, we can find that amplitudecoefficient b1 has a significant influence on amplitude coeffi-cients, Ap1;c, ap1;1, and bp1;1, for the core and periodic region.

This is because amplitude coefficient bm corresponding to theinward wave acts on the suppression of inward waves in theexternal region. In amplitude coefficients ap1;N�1 and a0p1;N�1of the external region, ηpa corresponding to the index changeis added to bm.

Remaining amplitude coefficients of the perturbed systemcan be calculated from the above amplitude coefficients withthe help of Eqs. (26), (27), (32), and (33). We are now in aposition that all the amplitude coefficients of the perturbedsystem are represented in terms of those of the unperturbedsystem. The amplitude coefficients of the perturbed system aswell as the unperturbed system are indispensable for the eva-luation of βp2 for general cases.

6. ANALYTICAL EXPRESSION OFPROPAGATION CONSTANT INPERTURBED SYSTEMThis section gives analytic expressions that are available forgeneral cases. Eθ component is used as Ψ in calculating inte-gral parameters.

A. Propagation Constant of First-Order PerturbationWhen Eθ;∞ is used asΨ∞ in Eq. (10), the propagation constantof the first-order perturbation can be expressed with thehelp of

Icore � 4π∣Ac∣2�ωμ0�2r2cκ2c

�J20 � J21 −

2κcrc

J0J1

�; �37�

Iex;a � 4�ωμ0�2κ4a

R2NTE1 − R2TE

f�∣a1∣2 � ∣b1∣2��κaa�

− ��a21 � b21� cos�κaa� − i�a21 − b21� sin�κaa�� sin�κaa�g;�38�

with RTE ≡ Re�XTE� � f�Re�XTE��2 − 1g1=2 (see Appendix A).The argument of Bessel function Jν is all κcrc in this section,and it is abbreviated for brevity. An expression of Iex;b can begiven where the symbol and subscript a are replaced by b, andthe prime is added to amplitude coefficients in Eq. (38). I∞aand I∞b are obtained by setting N � 0 in Iex;a and Iex;b, respec-tively. Itot is obtained using Eq. (11).

B. Propagation Constant of Second-Order PerturbationWe set Ψ∞ � Eθ;∞ and Ψp1 � Eθ;p1 in Eq. (12) to evaluateintegral parameters. Then, we can use Eq. (18) for the periodicregion and Eq. (34) for the external region, and employamplitude coefficients of the perturbed system derived inSection 5. Integral parameters are shown by neglecting termsin proportion to β∞βp1 in the following way. For the core, wehave

Ip;core � 4πA�c Ap1;c�ωμ0�2r2cκ2c

�J20 � J21 −

2κcrc

J0J1

�: �39�

For the periodic region a, we obtain


Ip;pe;a �4κa

�ωμ0κa

�2 1 − R2NTE1 − R2TE

�−�a�1ap1;1 � b�1bp1;1�a

� sin�κaa�κa�a�1bp1;1 exp�iκaa� � b�1ap1;1 exp�−iκaa��

�

�40�after summing up Ip;pe;a;m for the mth periodic layer a fromm � 1 to N . For the periodic region b, we have

Ip;pe;b �4κb

�ωμ0κb

�2 1−R2NTE1−R2TE

12

�κbκa

�1=2

×��ha11ap1;1�ha12bp1;1�

�−a0�1 b�b0�1

sin�κbb�κb

exp�−iκbb��

��ha21ap1;1�ha22bp1;1��−b0�1 b�a0�1

sin�κbb�κb

exp�iκbb��

�41�

in the similar way.For the external region a, one obtains

Ip;ex;a �4κa

�ωμ0κa

�2R2NTE

�−�1 − R2TE�−1

×��

1 −i2ηpaa

1� Cex exp�iKjΛ�1 − Cex exp�iKjΛ�

��a∣a1∣2

−sin�κaa�

κaa1b�1 exp�−iκaa�

�� a∣b1∣2

−sin�κaa�

κaa�1b1 exp�iκaa�

�

� 11 − Cex exp�−iKjΛ�

�aa�1 −

sin�κaa�κa

b�1 exp�−iκaa��

×�a1

�1 − iηpaa

Cex exp�iKjΛ�1 − Cex exp�iKjΛ�

�− b1

J�;aJ−;a

��42�

using a relation, exp�iK�j Λ� � exp�−iKjΛ�, derived from [17].The term ηpa represents the effect of index change in the ex-ternal layer a, and R2NTE reflects the dependence of number Nof cladding layer pairs in Eq. (42). Similarly, we have the in-tegral parameter for the external region b as

Ip;ex;b �4κb

�ωμ0κb

�2R2NTE

�−�1 − R2TE�−1

×��

1 −i2ηpbb

1� Cex exp�iKjΛ�1 − Cex exp�iKjΛ�

��b∣a01∣2

−sin�κbb�

κba01b

0�1 exp�−iκbb�

�� b∣b01∣2

−sin�κbb�

κba0�1 b

01 exp�iκbb�

�

� 11 − Cex exp�−iKjΛ�

12

�κbκa

�1=2

�1� κbκa


×�ba0�1 −

sin�κbb�κb

b0�1 exp�−iκbb��

×�a1�1� iηpaa� − b1

J�;aJ−;a

��: �43�

Both Ip;ex;a and Ip;ex;b include Cex, which reflects on the exis-tence of only the outward waves in the external region, unlikeother integral parameters.

All integral parameters in Eqs. (39)–(43) are represented interms of parameters of the unperturbed system and ηpi. Thepropagation constant βp2 of second-order perturbation is ob-tainable by substituting these integral parameters into Eq. (8).Thus, the CL for general cases can be computed.

7. EXPLICIT EXPRESSION OFPROPAGATION CONSTANT UNDER QWSCONDITIONThe QWS condition, κaa � κbb � π=2, has been introduced asa condition where an optical wave is efficiently confined tothe core in the Bragg fiber [5]. The QWS condition has beenextended to a generalized QWS condition [24]. Under the QWSand generalized QWS conditions, the eigenvalue equation ofthe Bragg fiber can be simplified [18,24]. Employment of sim-plified eigenvalue equations leads to an explicit expression ofthe propagation constant, as shown next. It provides severaluseful properties of CL.

A. Eigenvalue Equations under Generalized QWSConditionThe simplified eigenvalue equation of the TE mode in Braggfiber is given by [18,24]

κcrc � 2πrcλ0

�n2c −

�β∞k0

�2�1=2

� UQWS �44�

with UQWS � j1;μ � j00;μ�1 for the TE0μ mode. Here, jν;μ and j0ν;μindicate the μth zeros of Bessel function Jν and its derivativeJ 0ν with respect to the argument, respectively. ν and μ stand forthe azimuthal and radial mode numbers. Then, cladding thick-nesses a and b are determined by

aλ0

� q1 − 1=22

��n2a − n2c� �

�UQWSλ02πrc

�2�−1=2

; �45a�

bλ0

� q2 − q1 � 1=22

��n2b − n2c� �

�UQWSλ02πrc

�2�−1=2

; �45b�

where q1 and q2 are integers with q2 ≥ q1 ≥ 1 [24]. Equa-tions (45a) and (45b) consist of core, cladding, and mode para-meters. If we set q1 � q2 � 1 in Eqs. (45a) and (45b), then theabove expressions are reduced to the prior results under theQWS condition [18]. It is found from Eq. (45a) that claddingthickness a can be increased by increasing the value of q1 withthe remaining parameters unchanged.

Guided modes are allowed for jRe�XTE�j > 1 [17]. WhichBloch wavenumber K1 or K2 is chosen depends on the valueof Re�XTE�. We make use of K1 for Re�XTE� < −1, includingthe fundamental photonic band, and we offer explicit expres-sions under the QWS condition in this section.

B. Propagation Constant of First-Order PerturbationIntegral parameters relating to unperturbed fields are ob-tained to be

Itot � 4π∣Ac∣2�ωμ0�2J20�UQWS�

×r4c

U2QWS

�1� 1

n2a − n2b

Λrc

�UQWSrck0

�2�; �46�


Icore � 4π∣Ac∣2�ωμ0�2J20�UQWS�U2QWSκ4c

; �47�

Iex;a �1π ∣Ac∣

2�ωμ0�2J20�UQWS��ab

�2N rcaλ20

n2a − n2b: �48�

Expressions for Iex;b, I∞a, and I∞b are obtained in a mannersimilar to that given just after Eq. (38). It is natural that thevalue shown in Eq. (46) is real and positive.

The propagation constant of the first-order perturbationcan be given as

βp1k0

� − �a=b�2NaU2QWS

2�β∞=k0�r3ck20

�1� 1

n2a − n2b

Λrc

�UQWSrck0

�2�−1

�49�

by substituting Eqs. (46) and (48) into Eq. (10). Equation (49)brings out several properties, which are described later be-cause they are almost identical to βp2, namely, CL.

Let us estimate the magnitude of several important param-eters under the QWS condition. For the TE01 mode (UQWS �3:83171) with na � 2:5, nb � nex � 1:5, nc � 1:0, rc � 2:0 μm,and λ0 � 1:0 μm, we have β∞=k0 � 0:9524 from Eq. (44). In ad-dition, we obtain βp1=k0 � −2:63 × 10−3�0:1082=0:2157�2Nfrom Eq. (49). One obtains βp1=k0 � −2:64 × 10−6 and −2:60 ×10−9 for N � 5 and 10, respectively. Value of the second termin square brackets of Eq. (49) is 3:76 × 10−3, which is negligiblysmall compared to the unity for rc=λ0 ≫ 1.

We compare the value of �n2a − n2b�k20 with 2β∞βp1 in the ηpaof Eq. (21). The former is nearly of the same order as that of2β2∞ since (na � nb) is nearly of the same order as that of nc.Hence, the ratio of the former to the latter is roughly identicalto the ratio of β∞ to βp1. The fact of β∞ ≫ ∣βp1∣ indicates that�n2a − n2b�k20 is markedly larger than ∣2β∞βp1∣. For example,for the TE01 mode with the same parameters as the above,we have Rβ ≡ 2β∞βp1=��n2a − n2b�k20� � −1:253 × 10−3�0:1082=0:2157�2N . One obtains Rβ � −1:26 × 10−6 for N � 5 and−1:28 × 10−9 for N � 10.

C. Expression and Properties of CLThe βp2 can be calculated by substituting integral parameters,Eqs. (39)–(43), into Eq. (8). CL is obtained from the imaginarypart of the propagation constant. Im�Ip;ex;a� mainly contri-butes to the CL of the TE mode because �n2a − n2b�k20 ≫∣2β∞βp1∣. We show here only the results that apply toRe�XTE� < −1. An explicit expression of CL results in

Im�βp2�k0

�−�n2a−n2b�2�a=b�2NU2QWSa5k20

π3�β∞=k0�r3c×�1� 1

n2a−n2b

Λrc

�UQWSrck0

�2�−1 1

�b=4a��2�b=a�a=b�−1

×� �b=4a��2�b=a�a=b�1− �a=4b��2�b=a�a=b�

−1��b=4a��2�b=a�a=b�

2�1−�a=b�2�

��50�

under the generalized QWS condition.We can deduce several important properties from Eq. (50)

as follows.

i. Im�βp2� is always negative because the factor includinga=b is positive for 0 < a=b < 1, namely, na > nb.

ii. Im�βp2� includes the dependence of �a=b�2N on thenumber N of finite periodic layer pairs. Its absolute valueis reduced with an increase in N because 0 < a=b < 1.

iii. Im�βp2� is proportional to r−3c and U2QWS.iv. The last term is peculiar to the fact that only outward

waves exist in the external region.v. Moreover, the dependence of �n2a − n2b�2 is added to the

above result.vi. If fiber structural parameters, such as rc, a, and b, are

changed so as to satisfy the QWS condition for every wave-length λ0 at constant refractive indices, then the right-handside of Eq. (50) is independent of λ0. Hence, the CL valueis in inverse proportion to λ0. This means that the CL valueis small for a long wavelength when the QWS condition is al-ways maintained.

Items (i)–(iii) also hold for βp1, as can be seen from Eq. (49).The r−3c dependence of CL of the TEmode in Bragg fiber has

been shown by a consideration of the field form [7] and bynumerical examples for large rc=λ0 [12]. The agreement isreasonable because the asymptotic expansion approximationis valid for rc=λ0 ≫ 1. The �a=b�2N dependence of the TEmodehas been pointed out and confirmed by numerical examples[12]. The U2QWS dependence of CL is presented here for thefirst time. This means that higher-order modes exhibit alarge CL for fixed structural parameters, as confirmed inSubsection 8.A. It has been stated that the TE01 mode showsthe lowest CL among all the modes [7,12].

8. VERIFICATION OF CL BASED ONPRESENT THEORYThis section is devoted to the verification of CL derived fromthe present theory. The CL of the Bragg fiber has been calcu-lated by various methods, as stated in Section 1. It has numeri-cally been shown [12] that the MLD results are in excellentagreement with those of the transfer matrix and Chew’s meth-ods. Therefore, we use the results of the MLD method as thetouchstone of the CL values. Most fiber parameters treatedhere are set to meet the QWS condition for infinite claddinglayer pairs, even though the present paper is targeted forBragg fibers with finite cladding layer pairs.

In the following examples in this section, fiber parametersare always set to satisfy the QWS condition, namely, Eqs. (45a)and (45b) with q1 � q2 � 1 except for Fig. 6. Therefore,Eq. (50) is used to evaluate the CL for the present theory ex-cept for Fig. 2. The core index is assumed to be nc � 1:0throughout this paper.

A. Confirmation of Mode DependenceFigure 2 shows the normalized CL divided by U2QWS as a func-tion of core radius, where the CL of the TE0μ mode is redrawnfrom the previous result numerically evaluated by the MLDmethod [12]. As the core radius rc increases, the normalizedCL with identical na converges to a certain value despite ra-dial mode number μ. This supports the mode dependence de-scribed in item (iii) of Subsection 7.C, which is highly precise,especially for rc ≥ 7:0 μm. This agreement for large rc is due tothe fact that the asymptotic expansion approximation securesa high accuracy for large rc. The converging value depends on


the refractive index because the second term in square brack-ets of Eqs. (45a) and (45b) is negligibly small compared to itsfirst term, and hence both a and b are nearly independent of rcand the mode number.

B. Dependence on Number of Cladding Layer PairsThe dependence of CL on number N of cladding layer pairs issemilogarithmically shown in Fig. 3 for several TE0μ modes tocompare the results of the present theory with those of theMLDmethod. CL exhibits a nearly linear change withN . SlopeS for each mode is almost identical over the entire region,since CL is proportional to �a=b�2N for the TE mode [12].For example, the ratios of the values by the present at N �10 to those by the MLD method are 0.209, 0.226, and 0.250for TE01, TE02, and TE03 modes, respectively. This implies thatthe CL value of the present theory ranges from about one fifthto one fourth of that of the MLD method. CL is low for lower-order modes among the TE0μ modes, as expected from thevalue of UQWS in Eq. (50).

C. Dependence on Core RadiusThe core radius dependence of CL for the TE0μ modes is loga-rithmically plotted in Fig. 4 to compare the results of both

methods. CL gradually decreases against the core radius ex-cept for the small core radius, which is close to the guidinglimit. CL is inversely proportional to r3c for the large rc ofthe TE0μ modes, as seen from the present theory. r−3c depen-dence has been shown by a consideration of field form [7] andby numerical means [11]. The tendency of CL is nearly thesame for both methods, although the values of the present the-ory are about one fifth of those of the MLD method. For ex-ample, the ratios are 0.200, 0.201, and 0.202 for the TE01, TE02,and TE03 modes at rc � 10:0 μm.

D. Dependence on Cladding Index ContrastThe cladding index contrast dependence of CL is shown inFig. 5 for TE0μ modes with nb � nex � 1:5. CL is reduced withincreasing na. The discrepancy in the results between the pre-sent theory and MLD method increases with an increase in naand is nearly the same for a fixed value of na − nb, despitemode number μ. For example, we have loss ratios of 0.209,0.125, and 0.0860 at na � 2:5, 3.5, and 4.5 for the TE01 mode.

The difference in the results between the present theoryand MLD method is noteworthy for the cladding index con-trast compared to other fiber parameters. The ratios of thepresent theory to the MLD method are about one fifth, oneeighth, and one twelfth for na � 2:5, 3.5, and 4.5. The present

Fig. 2. (Color online) Normalized CL divided by U2QWS for severalTE0μ modes as a function of core radius: solid curve, TE01 mode;dashed curve, TE02 mode; dotted–dashed curve, TE03 mode.λ0 � 1:0 μm, nb � nex � 1:5, nc � 1:0, and N � 10. Cladding thick-nesses a and b are determined from Eqs. (45a) and (45b) withq1 � q2 � 1, namely, the QWS condition.

Fig. 3. (Color online) Comparison on cladding layer pairs depen-dence of CL for TE0μ mode. λ0 � 1:0 μm, rc � 2:0 μm, nc � 1:0, na �2:5, and nb � nex � 1:5. Cladding layer thicknesses are a � 0:1082 μmand b � 0:2157μm for TE01 mode, a � 0:1060μm and b � 0:2001 μmfor TE02 mode, and a � 0:1029 μm and b � 0:1811 μm for TE03 mode.They are determined from Eqs. (45a) and (45b) for each UQWS. Solidcurves, perturbation theory; dashed curves, MLD method.

Fig. 4. (Color online) Comparison of core radius dependence of CLfor TE0μ mode. λ0 � 1:0 μm, na � 2:5, nb � nex � 1:5, and N � 10.Cladding layer thicknesses a and b are determined from Eqs. (45a)and (45b) for each combination of rc, na, and nb. Solid curves, per-turbation theory; dashed curves, MLD method.

Fig. 5. (Color online) Comparison of cladding high-index depen-dence of CL for TE0μ mode. λ0 � 1:0 μm, rc � 2:0 μm, nb � nex �1:5, and N � 10. Cladding layer thicknesses a and b are determinedfrom Eqs. (45a) and (45b) for each combination of rc, na, and nb. Solidcurves, perturbation theory; dashed curves, MLD method.


theory is useful for evaluating CL, since it provides a simplecalculation method.

E. Dependence on WavelengthFigure 6 semilogarithmically indicates the wavelength depen-dence of CL for the TE0μ modes to compare the results of thepresent theory with those of MLD method. Fiber structureparameters are set to satisfy the QWS condition at λQWS �1:0 μm. After the fiber structure was fixed, the wavelengthwas varied. The loss is low for lower-order modes amongthe TE0μ modes and shows its minimum value in the neighbor-hood of the QWS condition. The loss discrepancy betweenboth methods is maximum near the QWS condition. CL valuesof the present theory are about one fifth and one fourth of theMLD method for the TE01 and TE03 modes, respectively, nearthe QWS condition.

Summarizing these results, the present theory is inclinedto underestimate the CL under the QWS condition. Thedifference in CL between both methods is noteworthy in thecladding high-index dependence. However, the present theoryenables us to compare CL values relatively.

9. RELATIONSHIP OF CL WITHGENERALIZED QWS CONDITIONThis section presents the numerical results of CL, mainly thecladding layer thickness dependence, and studies the relation-ship of CL characteristics with the generalized QWS condition.Hence, we exploit the CL expression for general cases. Thewavelength is fixed at λ0 � 1:0 μm in most cases.

A. Some Expressions under Generalized QWS ConditionThe relation between the photonic bandgap and antireso-nance was presented in the SPARROW model [13]. The in-phase condition is equivalent to a generalized QWS conditionin the 1D and cylindrically symmetric 2D structures with per-iodic cladding, and, moreover, the generalized QWS conditionis identical with the central gap point in the SPARROWmodel [24].

When the Bragg fiber exactly satisfies the generalized QWScondition of the cladding parameters, Eqs. (45a) and (45b),effective index β∞=k0 is given by [18,24]

~nco ��n2c −

�UQWSλ02πrc

�2�1=2

: �51�

Equation (51) includes the core and mode parameters.The generalized QWS condition, namely, the central gap

point, can also be represented using only cladding parametersin terms of the wavelength [13,24]:

λgQWS � 2�

1

n2a − n2b

��q1 − 1=2

a

�2−�q2 − q1 � 1=2

b

�2��−1=2

�52�

and the effective index:

~ncl ��n2a − n2bη2cl1 − η2cl

�1=2

�53�

with ηcl ≡ ��q1 − 1=2�b�=��q2 − q1 � 1=2�a�. The expressions inEqs. (52) and (53) agree exactly with those in [13] by settingq1 � m1 and q2 − q1 � m0.

B. Cladding Layer Thickness Dependence of CLAlthough cladding layer thickness b is set to satisfy the QWScondition in Eq. (45b) for nb, nc, and rc, for the moment, layerthickness a is arbitrarily changed. Figure 7 shows the a depen-dence of the CL of the TE01 mode as a function of core radiusrc. We show only the guided modes that satisfy jRe�XTE�j > 1[17]. Hence, the region without curves corresponds to the ra-diation mode. This peculiar property is attributed to the factthat the optical wave is confined to the core based on thephotonic bandgap mechanism in the Bragg fiber. Regions withfinite CL values are regarded as photonic bands. The CL forthe guided mode decreases with increasing rc at a fixed valueof a. We find, for any case, that CL exhibits its minimum valuenear the cross where cladding layer thickness ameets the gen-eralized QWS condition. CL is the lowest at q1 � q2 � 1,namely, the QWS condition. For example, the value of a is0:1061 μm at the lowest CL and is 0:1082 μm under the QWScondition for rc � 2:0 μm. Their relative difference in a isabout 2.0% for rc � 2:0 to 10:0 μm. Their CL values havealso nearly the same relative difference as in a. The valuesof a showing the minimum CL hardly depend on rc for each

Fig. 6. (Color online) Comparison of wavelength dependence of CLfor TE0μ mode. rc � 2:0 μm, na � 2:5, nb � nex � 1:5, and N � 10.Cladding layer thicknesses a and b are set to satisfy Eqs. (45a) and(45b) with q1 � q2 � 1 at λQWS � 1:0 μm and are the same as thosein Fig. 2 for TE01 to TE03 modes.

Fig. 7. (Color online) Dependence of CL on cladding layer thicknessa for TE01 mode as a function of core radius rc. λ0 � 1:0 μm, na � 2:5,nb � nex � 1:5, and N � 10. For rc � 2, 5, and 10 μm, values of clad-ding layer thickness b are 0.2157, 0.2223, and 0:2233 μm that satisfyEq. (45b) with q1 � q2 � 1. Crosses indicate positions at which thick-ness a satisfies the generalized QWS condition. Three crosses for eachrc correspond to q1 � 1, 2, and 3 from the left.


photonic band. As rc changes from 2.0 to 5:0 μm, CL is reducedby about 1.5 orders.

The dependence of CL on cladding layer thickness a isplotted in Fig. 8 for the TE01 mode as a function of claddinghigh-index na. We can also see the regions where no guidedmodes exist, as in Fig. 7. As na increases, the CL for the guidedmode greatly decreases and thickness a exhibiting the mini-mum loss shifts to a small value. There is a tendency that theminimum loss is achieved near the generalized QWS conditionand the lowest loss is obtained near the QWS condition: atq1 � q2 � 1. In addition, the spacing between the photonicbands becomes narrow with increasing na because the gener-alized QWS condition depends much more on the claddinglayer indices than on other parameters, as seen fromEqs. (45a)and (45b).

The a dependence of CL is shown in Fig. 9 for several TE0μmodes. CL is low for the TE01 mode and increases by aboutone order every radial mode number μ. The minimum loss canbe found at thickness a, where the generalized QWS conditionis satisfied. The lowest loss is also obtained near the QWS con-dition despite mode number μ.

Contrary to the above, cladding layer thickness b is arbitra-rily changed on the condition that thickness a is kept at theQWS condition. Figure 10 shows the CL versus b relation ofthe TE01 mode as a function of core radius rc. As rc increases,CL decreases and thickness b exhibiting the minimum loss isnearly unchanged. The minimum loss tends to be obtainednear the generalized QWS condition and the lowest loss is ob-tained near the QWS condition, as in the case of the a change.For example, the value of b is 0:2154 μm at the lowest CL andis 0:2157 μm under the QWS condition for rc � 2:0 μm. Theirrelative differences in b are 0.15 to 0.37% for rc � 2:0 to10:0 μm. The difference in cladding thickness for the QWScondition and the minimum CL is smaller in b than in a.

From Figs. 7–10, CL exhibits its minimum loss in the vici-nity of the generalized QWS condition and the lowest loss isobtained almost at the QWS condition if cladding layer thick-ness b or a satisfies the QWS condition. Cladding thickness ashowing the minimum CL is nearly independent of the coreradius and the radial mode number, but it strongly dependson the cladding index contrast.

The border between the guided and radiation modes is de-termined from the value of Re�XTE�: guided and radiation

modes satisfy jRe�XTE�j > 1 and jRe�XTE�j ≤ 1, respectively[17]. The contour curve of Re�XTE� � �1 is illustrated inFig. 11 for λ0 � 1:0 μm, rc � 2:0 μm, na � 2:5, and nb �nex � 1:5. Dotted–dashed lines indicate the position that satis-fies Eqs. (45a) and (45b), which is the generalized QWScondition. Cladding layer thicknesses a and b satisfy the gen-eralized QWS condition simultaneously at the crossing pointsof the dotted–dashed lines. Figure 11 shows that the crossingpoints are situated nearly at the center of the guided modearea, explaining why CL exhibits its minimum loss near thegeneralized QWS condition. A contour curve similar to Fig. 11was shown in the (κaa − κbb) plane of a previous work [25].

C. How to Find Operation Condition Exhibiting Low CLWe plot the wavelength dependence of CL of the TE01 mode inFig. 12 for point A in Fig. 11. The fiber parameters are set tosatisfy the QWS condition at λQWS � 1:0 μm. The results of theMLD method are redrawn from a previous paper [24]. We cansee good agreement of the results between the two methods.The loss values near 108 dB=km for N � 20 are attributed tothe computation limit of the MLD method. The present theoryshows that no CL values appear near λQWS=ℓ with even ℓ.

Fig. 8. (Color online) Dependence of CL on layer thickness a for theTE01 mode as function of cladding high-index na. λ0 � 1:0 μm,rc � 2:0 μm, nb � nex � 1:5, b � 0:2157 μm, and N � 10. Crosses in-dicate positions at which thickness a satisfies the generalized QWScondition. Three crosses for each na correspond to q1 � 1, 2, and3 from the left.

Fig. 9. (Color online) Dependence of CL on layer thickness a for theTE0μ mode as function of radial mode number μ. λ0 � 1:0 μm,rc � 2:0 μm, na � 2:5, nb � nex � 1:5, and N � 10. Values of claddinglayer thickness b are 0.2157, 0.2001, and 0:1811μm for TE01, TE02, andTE03 modes that meet Eq. (45b) with q1 � q2 � 1, namely, the QWScondition. Crosses indicate positions at which a satisfies the general-ized QWS condition. Three crosses for each mode correspond toq1 � 1, 2, and 3 from the left.

Fig. 10. (Color online) Dependence of CL on cladding layer thicknessb for the TE01 mode as a function of core radius rc. λ0 � 1:0 μm,na � 2:5, nb � nex � 1:5, and N � 10. Cladding layer thickness a isset to satisfy Eq. (45a) with q1 � 1. Crosses indicate positions at whichthickness b satisfies the generalized QWS condition. Three crosses foreach rc correspond to q1 � 1, 2, and 3 from the left.


Figure 12 shows that the fundamental photonic band ap-pears near λQWS � 1:0 μm. From Eqs. (52) and (53), λgQWS �1:0, 0.333, and 0:2 μm for three combinations of q1 � 1 andq2 − q1 � 0, q1 � 2, and q2 − q1 � 1, and q1 � 3 and q2 − q1 �2, respectively, and ~ncl � 0:9524 for three combinations. Onthe other hand, ~nco � 0:9524, 0.9948, and 0.9981 for the threecorresponding combinations despite N from Eq. (51). For thethree cases, both the present theory and MLD method provideidentical effective indices of 0.9524, 0.9948, and 0.9981 be-cause the values of ∣βp1=k0∣ are of the order of 10−6 and10−9 for N � 5 and 10. It is natural that Eq. (51) gives a preciseeffective index under the exact generalized QWS condition.This means that the core parameters are indispensable forthe precise estimation of the effective index.

Point B in Fig. 11 indicates where both q1 and q2 − q1 arehalf integers, namely, q1 � 1:5 and q2 − q1 � 0:5 in Eqs. (45a)and (45b), at λ0 � 1:0 μm. After the fiber structural parametersare fixed at point B, only the wavelength is varied (Fig. 13).The fiber has a guided region near λ0 � 2:0 μm. This can beexplained as follows. After substituting the cladding param-eters into Eq. (52), we have λgQWS � 2:000 μm for q1 � q2 � 1,λgQWS � 0:667 μm for q1 � 2 and q2 � 3, and λgQWS � 0:400 μmfor q1 � 3 and q2 � 5. Equation (53) gives an effective indexof ~ncl � 0:9524 for these three combinations, and Eq. (51)gives ~nco � 0:9925 at λ0 � 0:4 μm, ~nco � 0:9791 atλ0 � 0:667 μm, and ~nco � 0:7925 at λ0 � 2:0 μm. Actually,since we have β∞=k0 � 0:9964, 0.9790, and 0.7979 forλ0 � 0:4, 0.667, and 2:0 μm, respectively, we can estimateλgQWS but not the effective index from only the cladding para-meters, as in Fig. 12. Point B satisfies the QWS conditionat λQWS � 2:0 μm.

CL also shows its minimum loss at λ0 � 2:0=ℓ μmwith odd ℓ.The phase theory explained that CL shows a minimum loss atλQWS=ℓ with odd ℓ under the generalized QWS condition [24].This condition is equivalent to that in which waves reflected

from each cladding interface become in phase just inside thecore–cladding boundary of the Bragg fiber. Note from Fig. 13that no CL values appear near λ0 � 1:0 and 0:5 μm, whereguided modes exist. These wavelengths correspond to λQWS=ℓwith even ℓ. The present theory shows that the CL is too highto exhibit in the neighborhood of λQWS=ℓ with even ℓ.

Point C in Fig. 11 indicates the position where q1 is a halfinteger and q2 − q1 is an integer: q1 � 1:5 and q2 − q1 � 0 atλ0 � 1:0 μm. If we use the same procedure as in Fig. 12 to findλgQWS under a restriction of integers qi, we find them for acombination of large qis; that is, q1 � 5 and 7, for 0 < ~ncl < 1.

How do we estimate the wavelength of the photonic bandfor general cladding thicknesses? The width of the fundamen-tal photonic band is not so narrow. We seek a combination ofreal q1 and q2 near q1 � q2 � 1 such that the real solutions ofλgQWS, ~nco, and ~ncl exist simultaneously and the value of ∣ ~nco −~ncl∣ is as small as possible. For cladding indices and thick-nesses at point C, we seek solutions by changing q1 and q2around solution candidates every 0.01 steps. Then, we haveλgQWS � 1:486 μm, ~nco � 0:8915, and ~ncl � 0:8931 for q1 �1:18 and q2 � 1:03.

Figure 14 indicates the CL of the TE01 mode with fiberstructural parameters at point C. The fundamental photonic

Fig. 11. Contour curve of Re�XTE� in the �a; b� plane for the TE01mode. λ0 � 1:0 μm, rc � 2:0 μm, na � 2:5, and nb � nex � 1:5. Solidand dashed curves indicate Re�XTE� � −1 and �1. Guided modes ex-ist between solid curves or dashed curves. Dotted–dashed lines indi-cate cladding layer thicknesses a and b, which satisfy Eqs. (45a) and(45b) with integers q1 and q2. Point A corresponds to the position sa-tisfying the QWS condition, q1 � q2 � 1, at λ0 � 1:0 μm. Point B cor-responds to q1 � 1:5 and q2 − q1 � 0:5 at λ0 � 1:0 μm in Eqs. (45a) and(45b). Data for cladding thicknesses at point B are shown in Fig. 13.Point C indicates position for q1 � 1:5 and q2 − q1 � 0 at λ0 � 1:0 μm.Data for cladding thicknesses at point C are shown in Fig. 14.

Fig. 12. (Color online) Wavelength dependence of CL of the TE01mode for two kinds of N . Cladding thicknesses, a � 0:1082 μm andb � 0:2157 μm, are set at point A in Fig. 11. rc � 2:0 μm, na � 2:5,and nb � nex � 1:5. Solid curves, perturbation theory; dotted–dashedcurves, MLD method (N � 10); dashed curves, MLD method(N � 20). Crosses indicate data at wavelengths satisfying the general-ized QWS condition.

Fig. 13. (Color online) Wavelength dependence of CL for the TE01mode with cladding thicknesses at point B in Fig. 11. rc � 2:0 μm,na � 2:5, nb � nex � 1:5, a � 0:2163 μm, b � 0:4315 μm, andN � 10. Solid circles indicate data at wavelengths satisfying the gen-eralized QWS condition.


band is situated near λ0 � 1:55 μm, which is shifted from theabove λgQWS by about 4.1%. This slight discrepancy impliesthat the above estimation is reasonable. We have β∞=k0 �0:8966 at λ0 � 1:486 μm from the numerical data of the presenttheory. We cannot estimate the low CL regions using λQWS=ℓwith odd ℓ because there is no λQWS=ℓ in the present case. Thesecond and third photonic bands cannot be found in a mannersimilar to the above because their width is narrow. In the pre-sent case, far from the exact generalized QWS condition, wecan neither find a high loss near λ0 � 0:5 μm nor in the wave-length between the 0.75 and 1:55 μm bands.

It is possible to estimate the photonic band wavelength andthe effective index using Eqs. (52) and (51) when the Braggfiber approximately satisfies the generalized QWS condition.If the photonic band is set at a certain wavelength, we canemploy Eqs. (45a) and (45b) to design the fiber structuralparameters.

10. DISCUSSIONAlthough the present theory can predict the dependence of CLon fiber structural parameters, the number of finite periodiclayer pairs, wavelength, and mode number in an explicit form,it usually underestimates CL, as shown in Section 8. The rea-son is explored next.

CL is evaluated from the imaginary part Im�β� of the pro-pagation constant. Im�β� is extremely small compared to itsreal part Re�β�. The asymptotic expansion approximationand perturbation theory are used here to formulate CL. Ac-cording to the present theory, fields in external layer amainlycontribute to Im�β�, as shown in Section 7. The perturbation isgiven in the external region, where electromagnetic fields aremarkedly small in the unperturbed system. Amplitude coeffi-cients of the perturbed system have a strong dependence onamplitude coefficient b1, corresponding to the inward wave ofthe unperturbed system, as shown in Section 5. It is requiredfor CL to calculate perturbed fields as well as unperturbedfields with high precision. The asymptotic expansion approx-imation secures good accuracy for large rc=λ0, namely,κir ≫ 1. However, since perturbed fields are roughly obtainedby differentiating unperturbed fields, their accuracy is de-graded by the differentiation. The accuracy is insufficientto evaluate the subtle Im�β�, although it gives satisfactoryaccuracy for Re�β�. Therefore, the discrepancy between the

present theory and previous numerical results is attributedto the use of asymptotic expansion approximation.

11. CONCLUSIONAn analytical expression of CL was presented for the TEmodeof Bragg fibers using the perturbation theory. Under the gen-eralized QWS condition, the present theory leads to a formulaof CL that is proportional to �a=b�2Nr−3c U2QWS with the core ra-dius rc, the number N of finite periodic layer pairs, the zeroUQWS of the Bessel function peculiar to each mode, and clad-ding layer thicknesses a and b.

The present theory is useful for characterizations of CL, asseveral dependencies on the fiber structural parameters andthe wavelength. However, the present theory has a tendencyto underestimate CL. A significant difference can be seen inthe dependence of CL on cladding high-index na. Although thepresent theory usually underestimates CL, it provides deep in-sight into the PBF.

We scrutinized the relation between CL and the generalizedQWS condition. CL exhibits its minimum loss near the general-ized QWS condition. How to find the photonic band wasshown for general cases that do not satisfy the generalizedQWS condition.

APPENDIX A: SEVERAL PROPERTIESOF PERIODIC STRUCTURE UNDERASYMPTOTIC EXPANSIONAPPROXIMATIONFor the Bragg fiber with infinite periodic cladding, amplitudecoefficients of cladding layers a and b are related with eachother as [17,18]

a0∞;m � Haa∞;m; a∞;m�1 � Hba0∞;m �A1�

under the asymptotic expansion approximation. Here, matrixHa is identical to that in Eq. (24), and matrix elements Ha andHb are independent ofm. Usage of the Bloch theorem gives aneigenvalue of the product of matrices HbHa as

RTE ≡ exp�−iKjΛ� � Re�XTE� � f�Re�XTE��2 − 1g1=2 �A2�

with

XTE ��cos�κbb� −

i2

�κbκa

� κaκb

�sin�κbb�

�exp�−iκaa�: �A3�

Here, upper and lower signs correspond to j � 1; 2, and Blochwavenumbers K1 and K2 apply to Re�XTE� < −1 andRe�XTE� > 1, respectively. HbHa can be transformed into ascalar value exp�−iKjΛ� in the periodic region but not inthe external region. In addition, amplitude coefficients be-tween plural cladding layers a are related to each other as

a∞;m � exp�−iKjΛ�a∞;m−1 � exp�−iKj�m − 1�Λ�a∞;1: �A4�

Amplitude coefficients of cladding layers b are obtained byadding the prime to each a∞;m.

Fig. 14. (Color online) Wavelength dependence of CL for the TE01mode with cladding thicknesses at point C in Fig. 11. rc � 2:0 μm,na � 2:5, nb � nex � 1:5, a � 0:2163 μm, b � 0:2157 μm, andN � 10. Open circle indicates data at λgQWS � 1:486 μm.


APPENDIX B: AMPLITUDE COEFFICIENTSOF PERTURBED SYSTEMFrom Eq. (36a), we have the amplitude coefficient

Ap1;c

�πκarc2

�1=2

� 1J−;a

�−b1 � β∞βp1

rcκcAc

�πκarc2

�1=2

×�J 00 � i

κaκc

�J0 � 2J 00��

� iβ∞βp1�b1

Naκa

� 12Nbκb

exp�iKjΛ�

×�κaκb

�1=2

�−a01hb21 � b01hb22��

�B1�

for the core and

ap1;N�1exp�−iKjNΛ�

� −b1J�;aJ−;a

− ia1ηparNB

� iβ∞βp1�2Ac

�πκarc2

�1=2 κarc

κ2c�J0 � J 00�2

J−;a

� Naκa

�a1 � b1

J�;aJ−;a

�

� Nbκbexp�iKjΛ�

�κaκb

�3=2

�a01 exp�−iκbb�

J−;bJ−;a

� b01 exp�iκbb�J�;bJ−;a

��B2�

for the innermost external layer a, in a normalized form. Here,the argument κcrc of Jν and J 0ν is suppressed.

Use of Eq. (B1) leads to amplitude coefficients of innermostperiodic layer a as

ap1;1 � −b1J�;aJ−;a

� iβ∞βp1�−a1

rcκa

� b1Naκa

J�;aJ−;a

� 2Ac�πκarc

2

�1=2 κarc

κ2c�J0 � J 00�2

J−;a

� 12Nbκb

exp�iKjΛ��κaκb

�1=2

�−a01hb21 � b01hb22�J�;aJ−;a

�; �B3�

bp1;1 � −b1 � iβ∞βp1�b1

rc � Naκa

� 12Nbκb


�1=2

�−a01hb21 � b01hb22��: �B4�

Substituting Eq. (B2) into Eq. (33) with q � 1, we have theamplitude coefficient of innermost external layer b as

a0p1;N�1exp�−iKjNΛ�

�−a01�1� iηpbrN�1;A�

�12

�κbκa

�1=2

�1� κbκa


×�a1 −b1

J�;aJ−;a

� ia1ηpaa

� iβ∞βp1�2Ac

�πκarc2

�1=2 κarc

κ2c�J0�J 00�2

J−;a

�Naκa

�a1�b1

J�;aJ−;a

��Nbκb


�3=2

× �a01exp�−iκbb��b01exp�iκbb��J−;bJ−;a

��: �B5�

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analytical expression of confinement loss in bragg fibers and ......the antiresonant reflecting...

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